The logarithmic perturbation theory for bound states in spherical-symmetric potentials via the $\hbar$-expansions

TL;DR

This paper develops a semiclassical logarithmic perturbation theory using $ abla$-expansions for bound states in spherical potentials, providing a new recursive method for energy calculations in quantum systems.

Contribution

It introduces a novel perturbation expansion technique based on $ abla$-expansions, improving upon standard methods for spherical anharmonic and screened Coulomb potentials.

Findings

Derived new recursive formulas for energy eigenvalues.

Applied method successfully to quartic anharmonic oscillator.

Extended approach to Debye potential energy calculations.

Abstract

The explicit semiclassical treatment of the logarithmic perturbation theory for the bound-state problem for the spherical anharmonic oscillator and the screened Coulomb potential is developed. Based upon the $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As examples, the perturbation expansions for the energy eigenvalues of the quartic anharmonic oscillator and the Debye potential are considered.